I know that one can have a greater or lesser amount of surety (i.e. "I'm not really sure"), but don't you either know or not know something? Are there degrees of knowledge? I hear this phrase often from reporters and news correspondents: "We don't really know..." I think that this is a way to avoid saying "We don't know", which sounds more blunt, but is more precise.
Frequently, it is simply a matter of preference. However, the following variations could be interpreted slightly differently.
I don't know.
A simple declaration. Unambiguous.
I don't really know.
This is often used to deflect a negative self-assessment. It's common to not want to admit that we don't know something, so we occasionally de-emphasize the declaration.
I really don't know.
You didn't mention this variation, but this is often used to do the opposite, actually emphasizing the fact that we don't know. This can be used to stress to someone who may not believe you, that you in fact, really don't know something.
These are really just minor nuances, and the phrases are often used interchangeably. If someone wants to express a degree of knowledge, some other phrase is usually used in conjunction with the declaration. For example, "I don't know enough about X to answer that."
I don't really know suggests that the speaker may know a little about the subject discussed but not really that much.
I don't know Jimi
probably means you have never heard of Jimi, while
I don't really know Jimi
means you may have heard of or even met Jimi but you do not know much about him.
Different languages emphasize different things. Japanese, for example, has a tendency to emphasize the speaker's belief in the probability of events and the certainty of knowledge. The number of ways a Japanese speaker can say "I think it will rain tomorrow" is quite large.
English, in contrast, is very sensitive to order in which things occur or are likely to occur, as evidenced by such things as the future perfect tense (e.g., "by then I will have known the result).
But English also has a strong bias toward stress, so that degrees of knowledge are able to be pinpointed: I don't know. I'm not sure. I sort of know. I guess. Maybe. It's possible that... Etc.
The word really used in this context is an all-purpose intensifier:
I really don't understand.
You're really getting on my nerves.
She really had me going there for a while.
In each of those cases the really could be dropped without a change of meaning, but the level of intensity of the statement would not be the same. The word fairly implores the listener to believe what is being said. It anticipates and attempts to preempt objections to a statement. In the normal course of a conversation, such an objection might be raised as the word itself.
John: I don't understand.
John: Yeah, really. I just don't get it.
By adding the really John would be attempting to avoid having to repeat himself.
To my mind, I'm not really sure, suggests that the speaker might not be certain of something, but they have some supporting evidence to think so. For example:
Why did he leave?
I don't really know. However, I suspect it had something to do with the waffles.
Rather than hedging on the side of not knowing something, it can be hedging in the other direction.